# Poisson: Create a Poisson distribution In distributions3: Probability Distributions as S3 Objects

## Description

Poisson distributions are frequently used to model counts.

## Usage

 `1` ```Poisson(lambda) ```

## Arguments

 `lambda` The shape parameter, which is also the mean and the variance of the distribution. Can be any positive number.

## Details

We recommend reading this documentation on https://alexpghayes.github.io/distributions3/, where the math will render with additional detail.

In the following, let X be a Poisson random variable with parameter `lambda` = λ.

Support: {0, 1, 2, 3, ...}

Mean: λ

Variance: λ

Probability mass function (p.m.f):

P(X = k) = λ^k e^(-λ) / k!

Cumulative distribution function (c.d.f):

P(X ≤ k) = e^(-λ) ∑_{i = 0}^k λ^i / i!

Moment generating function (m.g.f):

E(e^(tX)) = e^(λ (e^t - 1))

## Value

A `Poisson` object.

Other discrete distributions: `Bernoulli()`, `Binomial()`, `Categorical()`, `Geometric()`, `HyperGeometric()`, `Multinomial()`, `NegativeBinomial()`
 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15``` ```set.seed(27) X <- Poisson(2) X random(X, 10) pdf(X, 2) log_pdf(X, 2) cdf(X, 4) quantile(X, 0.7) cdf(X, quantile(X, 0.7)) quantile(X, cdf(X, 7)) ```